Delta-Complete Decision Procedures for Satisfiability over the Reals

We introduce the notion of "\delta-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational number \delta, a \delta-complete decision procedure determines either that \varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that allows \delta-bounded numerical perturbations on \varphi. We prove the existence of \delta-complete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded \delta-SMT problem is in NP^C. \delta-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL framework, which integrates Interval Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient conditions for its \delta-completeness. We discuss practical applications of \delta-complete decision procedures for correctness-critical applications including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201

Proof Generation from Delta-Decisions

We show how to generate and validate logical proofs of unsatisfiability from delta-complete decision procedures that rely on error-prone numerical algorithms. Solving this problem is important for ensuring correctness of the decision procedures. At the same time, it is a new approach for automated theorem proving over real numbers. We design a first-order calculus, and transform the computational steps of constraint solving into logic proofs, which are then validated using proof-checking algorithms. As an application, we demonstrate how proofs generated from our solver can establish many nonlinear lemmas in the the formal proof of the Kepler Conjecture.Comment: Appeared in SYNASC'1

Satisfiability Modulo ODEs

We study SMT problems over the reals containing ordinary differential equations. They are important for formal verification of realistic hybrid systems and embedded software. We develop delta-complete algorithms for SMT formulas that are purely existentially quantified, as well as exists-forall formulas whose universal quantification is restricted to the time variables. We demonstrate scalability of the algorithms, as implemented in our open-source solver dReal, on SMT benchmarks with several hundred nonlinear ODEs and variables.Comment: Published in FMCAD 201

Revisiting the Complexity of Stability of Continuous and Hybrid Systems

We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using first-order formulas over the real numbers, and reduce stability problems to the delta-decision problems of these formulas. The framework allows us to obtain a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the stability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we give upper bounds on their degrees of unsolvability

Quantifier Elimination over Finite Fields Using Gr\"obner Bases

We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gr\"obner basis methods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite fields. We analyze the theoretical complexity of the algorithm and show its application in the formal analysis of a biological controller model.Comment: A shorter version is to appear in International Conference on Algebraic Informatics 201

Wide frequency tuning of continuous terahertz wave generated by difference frequency mixing under exciton-excitation conditions in a GaAs/AlAs multiple quantum well

Continuous terahertz wave sources with narrow bandwidth and wide frequency tunability enable high-resolution terahertz spectroscopy and high-speed information communication. In this study, using the optical nonlinearity of excitons as the source of second-order nonlinear polarization, we realize a continuous terahertz electromagnetic wave demonstrating wide frequency tunability from 0.1 to 18 THz without a decrease in intensity due to phonon scattering. Because of excitation of two exciton states in a Ga As / Al As multiple quantum well using two continuous-wave lasers, terahertz waves are emitted as a result of difference-frequency mixing, where the intensity shows a square dependence on the excitation intensity. Using the inhomogeneous width of exciton lines, we achieve wide frequency tunability without phonon effects

Learning Probabilistic Systems from Tree Samples

We consider the problem of learning a non-deterministic probabilistic system consistent with a given finite set of positive and negative tree samples. Consistency is defined with respect to strong simulation conformance. We propose learning algorithms that use traditional and a new "stochastic" state-space partitioning, the latter resulting in the minimum number of states. We then use them to solve the problem of "active learning", that uses a knowledgeable teacher to generate samples as counterexamples to simulation equivalence queries. We show that the problem is undecidable in general, but that it becomes decidable under a suitable condition on the teacher which comes naturally from the way samples are generated from failed simulation checks. The latter problem is shown to be undecidable if we impose an additional condition on the learner to always conjecture a "minimum state" hypothesis. We therefore propose a semi-algorithm using stochastic partitions. Finally, we apply the proposed (semi-) algorithms to infer intermediate assumptions in an automated assume-guarantee verification framework for probabilistic systems.Comment: 14 pages, conference paper with full proof